Sum Sn = 2 * 1 / 3 + 4 * 1 / 9 + 6 * 1 / 27 + (2n + 1 / 3 ^ n) It's two times one-third, not two times one divided by three

Offset subtraction
Take the original form as one
Then multiply each term of the original formula by one-third as formula 2
Then the subtraction of Formula 1 and formula 2 will result in an equal ratio sequence. Just use the summation formula

Sum Sn = 1-3 + 5-7 +... + (- 1) ^ (n-1) (2n-1)

Sn=(-2)+(-2)+…… N is even
Sn=1+2+2+2+…… N is odd
So Sn = (- 2) * n / 2 even
1 + 2 * (n-1) / 2 odd
The answer is
Sn = - N, n is even
Sn = n, n is odd

The formula of split term elimination method should be complete

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